A small company can produce up to 200 handmade sandals a month. The monthly cost, C, of producing x sandals is C(x) = 1000 + 5x. The monthly revenue, R, is given by R(x) = 75x - 0.4x2
a. For what values of x is the revenue greater than the cost?
b. At what production levels will the company make a profit?
c. At what production levels will the company lose money?
(a) when R(x) > C(x) -- so just plug in their definitions and solve the inequality.
75x - 0.4x^2 > 1000+5x
16 < x < 159
(b) exactly the same question as (a)
(c) everywhere else than in (b)
a. When the revenue is greater than the cost, we have R(x) > C(x). Substituting the given equations, we get:
75x - 0.4x^2 > 1000 + 5x
Simplifying:
-0.4x^2 + 70x - 1000 > 0
To solve this quadratic inequality, we can either use the quadratic formula, factoring, or graphing. Factoring is not straightforward in this case, so let's use the quadratic formula to solve for x:
x = [-70 ± √(70^2 - 4(-0.4)(-1000))] / (2(-0.4))
Simplifying:
x = [-70 ± √(4900 - 1600)] / (-0.8)
x = [-70 ± √3300] / (-0.8)
Now we have two potential solutions for x, let's calculate them separately:
x = (-70 + √3300) / (-0.8) ≈ -30.41 (ignore this negative solution for the production level)
x = (-70 - √3300) / (-0.8) ≈ 45.41
Therefore, the revenue is greater than the cost for production levels x > 45.41.
b. To find the production levels at which the company makes a profit, we need to consider when R(x) > C(x) and C(x) < 0 (cost is negative). Substituting the given equations, we get:
75x - 0.4x^2 > 1000 + 5x (Revenue > Cost)
1000 + 5x < 0 (Cost < 0)
Simplifying the second inequality:
5x < -1000
x < -200
Since production levels cannot be negative, the company does not make any profit.
c. To find the production levels at which the company loses money, we need to consider when the cost is greater than the revenue, i.e., C(x) > R(x). Substituting the given equations, we get:
1000 + 5x > 75x - 0.4x^2
Shifting all the terms to one side:
0.4x^2 - 70x + 1000 > 0
Now, we can use the quadratic formula to solve for x:
x = [-(-70) ± √((-70)^2 - 4(0.4)(1000))] / (2(0.4))
Simplifying:
x = [70 ± √(4900 - 1600)] / (0.8)
x = [70 ± √3300] / 0.8
Now we have two potential solutions for x, let's calculate them separately:
x = (70 + √3300) / 0.8 ≈ 123.14
x = (70 - √3300) / 0.8 ≈ 96.86
Therefore, the company loses money when producing between approximately 96.86 and 123.14 sandals.
To determine when the revenue is greater than the cost, we need to compare the revenue and cost equations and find the values of x that satisfy the inequality R(x) > C(x).
a. Revenue greater than cost:
R(x) > C(x)
75x - 0.4x^2 > 1000 + 5x
To simplify this inequality, we move all terms to one side:
0.4x^2 - 70x + 1000 > 0
Next, we can solve for x by factoring or using the quadratic formula. However, in this case, it is easier to find the x-values graphically or by using a calculator.
Using a graphing calculator or an online graphing tool, graph the function f(x) = 0.4x^2 - 70x + 1000.
The inequality 0.4x^2 - 70x + 1000 > 0 represents the area above the x-axis in the graph. Therefore, the solution is the set of x-values when the graph is above the x-axis.
From the graph, we can see that the revenue is greater than the cost when x is between approximately 83.7 and 151.3. Thus, the values of x for which the revenue is greater than the cost are x > 83.7 and x < 151.3.
b. Profit:
The company makes a profit when the revenue is greater than or equal to the cost, so we need to find the production levels at which R(x) ≥ C(x).
R(x) ≥ C(x)
75x - 0.4x^2 ≥ 1000 + 5x
Rearranging the inequality:
0.4x^2 - 70x + 1000 ≤ 0
Again, we can find the x-values by graphing or using a calculator.
From the graph, we can see that the profit is made when x is between approximately 0 and 83.7, and when x is between approximately 151.3 and 200. Thus, the production levels at which the company makes a profit are 0 ≤ x ≤ 83.7 and 151.3 ≤ x ≤ 200.
c. Loss:
The company will lose money when the cost is greater than the revenue, so we need to find the production levels at which C(x) > R(x).
C(x) > R(x)
1000 + 5x > 75x - 0.4x^2
Rearranging the inequality:
0.4x^2 - 70x + 1000 < 0
Again, we can find the x-values by graphing or using a calculator.
From the graph, we can see that the company will lose money when x is between approximately 83.7 and 151.3. Thus, the production levels at which the company loses money are 83.7 < x < 151.3.
To determine the values of x where the revenue is greater than the cost, we need to compare the revenue function R(x) with the cost function C(x) and find the values of x that satisfy the inequality R(x) > C(x).
a. Revenue > Cost
R(x) > C(x)
75x - 0.4x^2 > 1000 + 5x
To simplify the inequality, we can rearrange the terms:
0.4x^2 - 70x + 1000 < 0
Now, we have a quadratic inequality. To solve it, we can use the quadratic formula or graph the quadratic equation and find the values where the parabola is below the x-axis.
The quadratic formula is:
x = (-b ± √(b^2 - 4ac)) / (2a)
In this case, a = 0.4, b = -70, and c = 1000.
Using the quadratic formula, we get:
x = (70 ± √(70^2 - 4(0.4)(1000))) / (2 * 0.4)
Simplifying further:
x = (70 ± √(4900 - 1600)) / 0.8
x = (70 ± √3300) / 0.8
Approximating the square root of 3300 to be around 57.45, we have two possible solutions:
x ≈ (70 + 57.45) / 0.8 ≈ 183.06
x ≈ (70 - 57.45) / 0.8 ≈ 16.94
The company will generate revenue greater than the cost when producing between approximately 16.94 and 183.06 sandals per month.
b. To find the production levels at which the company makes a profit, we need to determine the values of x where the revenue is greater than the cost, that is R(x) > C(x).
Using the inequality found in part a:
75x - 0.4x^2 > 1000 + 5x
Simplifying the inequality:
0.4x^2 - 70x + 1000 < 0
The company will make a profit when producing between approximately 16.94 and 183.06 sandals per month.
c. To find the production levels at which the company loses money, we need to determine the values of x where the revenue is less than the cost, that is R(x) < C(x).
Using the inequality:
75x - 0.4x^2 < 1000 + 5x
Simplifying the inequality:
0.4x^2 - 70x + 1000 > 0
The company will lose money when producing fewer than approximately 16.94 or more than 183.06 sandals per month.